the potential-sum is less than 1. Hence: If the initial sum is more than 0, and Maker can play such that the potential-sum weakly increases, then this is a winning strategy for Maker; If the initial sum is less than 1, and Breaker can play such that the potential-sum weakly decreases, then this is a winning strategy for Breaker. A winning condition for Breaker Paul Erdős and John Selfridge presented a general condition that guarantees Breaker a winning strategy. They used a potential-based strategy. They defined the potential of any (non-broken) winning-set with unoccupied vertices as . So the potential of a set occupied by Maker is indeed . Whenever Maker takes an element, the potential of every set containing it increases to , i.e., increases by ; whenever Breaker takes an element, the potential of every set containing it drops to 0, i.e., decreases by . To every element, we assign a value which equals the total potential-increase in case Maker takes it, i.e., . The winning strategy of Breaker is pick an element with a highest value. This guarantees that, from the first turn of Breaker onwards, the potential always weakly decreases. Hence, if the potential at the first Breaker turn is less than 1, Breaker wins. In Maker's first turn, he can, at most, double the potential (by taking an element contained in all winning-sets). Therefore, it is sufficient that, at the game start, the potential is less than 1/2. To summarize, the Erdős-Selfridge theorem says that: If , then is Breaker's win. The theorem gives a very easy-to-check condition, and when this condition is satisfied, it also gives an efficient algorithm for computing Breaker's optimal strategy. The potential function has a probabilistic interpretation. The potential of a winning-set is the probability that, if the game is played randomly from now on, Maker will own that set. The potential-sum is thus the expected number of winning-sets owned by Maker if the game is played randomly. Whenever the potential-sum is less than 1, there must be a way to play the game such that the number of sets owned by Maker is 0. By ensuring that the potential-sum remains below 1, Breaker essentially de-randomizes this probabilistic claim until at the end of the game, it becomes a certainty. Note that if Breaker plays first, the condition changes to . In particular, if the winning-sets are all of size